Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 100.0%

Time: 9.5s

Alternatives: 18

Speedup: 2.2×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Initial Program: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma (* (+ J J) (sinh l)) (cos (* -0.5 K)) U))

C

double code(double J, double l, double K, double U) {
	return fma(((J + J) * sinh(l)), cos((-0.5 * K)), U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(Float64(J + J) * sinh(l)), cos(Float64(-0.5 * K)), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 87.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

   3. lower-fma.f64 87.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

   5. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

   6. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

   7. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

   8. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

   9. sinh-undef N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

   13. lower-sinh.f64 100.0

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

   14. lift-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

   15. cos-neg-rev N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

   16. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

   17. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

   18. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

   19. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

   20. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

5. Taylor expanded in K around inf

   \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

   3. count-2-rev N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

   4. lower-+.f64 100.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

9. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]
10. Add Preprocessing

Alternative 2: 46.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+21} \lor \neg \left(t\_0 \leq 10^{+79}\right):\\ \;\;\;\;\left(\ell \cdot J\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))))
   (if (or (<= t_0 -2e+21) (not (<= t_0 1e+79))) (* (* l J) 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double tmp;
	if ((t_0 <= -2e+21) || !(t_0 <= 1e+79)) {
		tmp = (l * J) * 2.0;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (exp(l) - exp(-l))
    if ((t_0 <= (-2d+21)) .or. (.not. (t_0 <= 1d+79))) then
        tmp = (l * j) * 2.0d0
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (Math.exp(l) - Math.exp(-l));
	double tmp;
	if ((t_0 <= -2e+21) || !(t_0 <= 1e+79)) {
		tmp = (l * J) * 2.0;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (math.exp(l) - math.exp(-l))
	tmp = 0
	if (t_0 <= -2e+21) or not (t_0 <= 1e+79):
		tmp = (l * J) * 2.0
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	tmp = 0.0
	if ((t_0 <= -2e+21) || !(t_0 <= 1e+79))
		tmp = Float64(Float64(l * J) * 2.0);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (exp(l) - exp(-l));
	tmp = 0.0;
	if ((t_0 <= -2e+21) || ~((t_0 <= 1e+79)))
		tmp = (l * J) * 2.0;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+21], N[Not[LessEqual[t$95$0, 1e+79]], $MachinePrecision]], N[(N[(l * J), $MachinePrecision] * 2.0), $MachinePrecision], U]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -2e21 or 9.99999999999999967e78 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 98.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 98.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot 2, \cos \left(-0.5 \cdot K\right), U\right)} \]

   8. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 23.4%

       \[\leadsto \mathsf{fma}\left(\ell \cdot J, \color{blue}{2}, U\right) \]

   11. Taylor expanded in J around inf

       \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\ell}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 23.3%

       \[\leadsto \left(\ell \cdot J\right) \cdot 2 \]

   if -2e21 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 9.99999999999999967e78

   1. Initial program 75.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Applied rewrites 74.3%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq -2 \cdot 10^{+21} \lor \neg \left(J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq 10^{+79}\right):\\ \;\;\;\;\left(\ell \cdot J\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.84:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \mathsf{fma}\left(K \cdot K, -0.125, 1\right), U\right)\\ \mathbf{elif}\;t\_0 \leq -0.03:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.84)
     (fma (* (* J 2.0) (sinh l)) (fma (* K K) -0.125 1.0) U)
     (if (<= t_0 -0.03)
       (fma (* (+ J J) l) (cos (* -0.5 K)) U)
       (fma (* (+ J J) (sinh l)) 1.0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.84) {
		tmp = fma(((J * 2.0) * sinh(l)), fma((K * K), -0.125, 1.0), U);
	} else if (t_0 <= -0.03) {
		tmp = fma(((J + J) * l), cos((-0.5 * K)), U);
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.84)
		tmp = fma(Float64(Float64(J * 2.0) * sinh(l)), fma(Float64(K * K), -0.125, 1.0), U);
	elseif (t_0 <= -0.03)
		tmp = fma(Float64(Float64(J + J) * l), cos(Float64(-0.5 * K)), U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.84], N[(N[(N[(J * 2.0), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, -0.03], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.839999999999999969

   1. Initial program 96.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 96.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{1 + \frac{-1}{8} \cdot {K}^{2}}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 71.9%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}, U\right) \]

   if -0.839999999999999969 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.029999999999999999

   1. Initial program 86.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 86.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 99.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 99.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in l around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \color{blue}{\ell}, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 72.4%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \color{blue}{\ell}, \cos \left(-0.5 \cdot K\right), U\right) \]

   if -0.029999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 85.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 96.2%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.845:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.845)
     (+
      (*
       (*
        J
        (*
         (fma
          (fma 0.016666666666666666 (* l l) 0.3333333333333333)
          (* l l)
          2.0)
         l))
       t_0)
      U)
     (fma (* (+ J J) (sinh l)) 1.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.845) {
		tmp = ((J * (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.845)
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * t_0) + U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.845], N[(N[(N[(J * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.84499999999999997

   1. Initial program 89.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

   5. Applied rewrites 93.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 0.84499999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 85.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 98.4%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.845:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.845)
   (fma
    (*
     (+ J J)
     (*
      (fma (fma 0.008333333333333333 (* l l) 0.16666666666666666) (* l l) 1.0)
      l))
    (cos (* -0.5 K))
    U)
   (fma (* (+ J J) (sinh l)) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.845) {
		tmp = fma(((J + J) * (fma(fma(0.008333333333333333, (l * l), 0.16666666666666666), (l * l), 1.0) * l)), cos((-0.5 * K)), U);
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.845)
		tmp = fma(Float64(Float64(J + J) * Float64(fma(fma(0.008333333333333333, Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l)), cos(Float64(-0.5 * K)), U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.845], N[(N[(N[(J + J), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.84499999999999997

   1. Initial program 89.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 89.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 99.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 99.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in l around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {\ell}^{2}\right)\right)\right)}, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 93.6%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right)}, \cos \left(-0.5 \cdot K\right), U\right) \]

   if 0.84499999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 85.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 98.4%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.48:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.48)
     (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) t_0) U)
     (fma (* (+ J J) (sinh l)) 1.0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.48) {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * t_0) + U;
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.48)
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * t_0) + U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.48], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.47999999999999998

   1. Initial program 89.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

   5. Applied rewrites 92.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 0.47999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 86.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 97.7%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.48:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.48)
   (fma (* (cos (* -0.5 K)) (* J (fma (* l l) 0.3333333333333333 2.0))) l U)
   (fma (* (+ J J) (sinh l)) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.48) {
		tmp = fma((cos((-0.5 * K)) * (J * fma((l * l), 0.3333333333333333, 2.0))), l, U);
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.48)
		tmp = fma(Float64(cos(Float64(-0.5 * K)) * Float64(J * fma(Float64(l * l), 0.3333333333333333, 2.0))), l, U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.48], N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.47999999999999998

   1. Initial program 89.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   if 0.47999999999999998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 86.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 97.7%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 87.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.03:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.03)
   (fma (* (+ J J) l) (cos (* -0.5 K)) U)
   (fma (* (+ J J) (sinh l)) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.03) {
		tmp = fma(((J + J) * l), cos((-0.5 * K)), U);
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.03)
		tmp = fma(Float64(Float64(J + J) * l), cos(Float64(-0.5 * K)), U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.03], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.029999999999999999

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 91.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 99.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 99.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in l around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \color{blue}{\ell}, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 64.9%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \color{blue}{\ell}, \cos \left(-0.5 \cdot K\right), U\right) \]

   if -0.029999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 85.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 96.2%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 86.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.03:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.03)
   (fma
    (* (fma (* K K) -0.125 1.0) (* J (fma (* l l) 0.3333333333333333 2.0)))
    l
    U)
   (fma (* (+ J J) (sinh l)) 1.0 U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.03) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (J * fma((l * l), 0.3333333333333333, 2.0))), l, U);
	} else {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.03)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(J * fma(Float64(l * l), 0.3333333333333333, 2.0))), l, U);
	else
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.03], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(J * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.029999999999999999

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]

   5. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right)\right), \ell, U\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right) \]

   if -0.029999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 85.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 96.2%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 95.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  (*
   (*
    J
    (*
     (fma
      (fma
       (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
       (* l l)
       0.3333333333333333)
      (* l l)
      2.0)
     l))
   (cos (/ K 2.0)))
  U))

C

double code(double J, double l, double K, double U) {
	return ((J * (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
}

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 87.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

5. Applied rewrites 96.9%

   \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
6. Add Preprocessing

Alternative 11: 95.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(J + J\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right), \cos \left(-0.5 \cdot K\right), U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma
  (*
   (+ J J)
   (*
    (fma
     (fma
      (fma 0.0001984126984126984 (* l l) 0.008333333333333333)
      (* l l)
      0.16666666666666666)
     (* l l)
     1.0)
    l))
  (cos (* -0.5 K))
  U))

C

double code(double J, double l, double K, double U) {
	return fma(((J + J) * (fma(fma(fma(0.0001984126984126984, (l * l), 0.008333333333333333), (l * l), 0.16666666666666666), (l * l), 1.0) * l)), cos((-0.5 * K)), U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(Float64(J + J) * Float64(fma(fma(fma(0.0001984126984126984, Float64(l * l), 0.008333333333333333), Float64(l * l), 0.16666666666666666), Float64(l * l), 1.0) * l)), cos(Float64(-0.5 * K)), U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J + J), $MachinePrecision] * N[(N[(N[(N[(0.0001984126984126984 * N[(l * l), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 87.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

   3. lower-fma.f64 87.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

   5. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

   6. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

   7. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

   8. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

   9. sinh-undef N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

   13. lower-sinh.f64 100.0

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

   14. lift-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

   15. cos-neg-rev N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

   16. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

   17. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

   18. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

   19. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

   20. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

5. Taylor expanded in K around inf

   \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

   3. count-2-rev N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

   4. lower-+.f64 100.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

9. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

10. Taylor expanded in l around 0

    \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \color{blue}{\left(\ell \cdot \left(1 + {\ell}^{2} \cdot \left(\frac{1}{6} + {\ell}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {\ell}^{2}\right)\right)\right)\right)}, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]
11. Step-by-step derivation

12. Applied rewrites 96.9%

    \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, \ell \cdot \ell, 0.008333333333333333\right), \ell \cdot \ell, 0.16666666666666666\right), \ell \cdot \ell, 1\right) \cdot \ell\right)}, \cos \left(-0.5 \cdot K\right), U\right) \]
13. Add Preprocessing

Alternative 12: 77.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.03:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(J \cdot t\_0\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (fma (* l l) 0.3333333333333333 2.0)))
   (if (<= (cos (/ K 2.0)) -0.03)
     (fma (* (fma (* K K) -0.125 1.0) (* J t_0)) l U)
     (fma (* t_0 l) J U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = fma((l * l), 0.3333333333333333, 2.0);
	double tmp;
	if (cos((K / 2.0)) <= -0.03) {
		tmp = fma((fma((K * K), -0.125, 1.0) * (J * t_0)), l, U);
	} else {
		tmp = fma((t_0 * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = fma(Float64(l * l), 0.3333333333333333, 2.0)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.03)
		tmp = fma(Float64(fma(Float64(K * K), -0.125, 1.0) * Float64(J * t_0)), l, U);
	else
		tmp = fma(Float64(t_0 * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.03], N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(t$95$0 * l), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.029999999999999999

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]

   5. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right)\right), \ell, U\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right) \]

   if -0.029999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]

   5. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, \color{blue}{J}, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 91.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;\ell \leq -780:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.00035:\\ \;\;\;\;\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right) + U\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (* (* (* (fma (* l l) 0.3333333333333333 2.0) l) J) (cos (* 0.5 K)))))
   (if (<= l -780.0)
     t_0
     (if (<= l 0.00035)
       (+ (* (* (* 2.0 J) l) (cos (* -0.5 K))) U)
       (if (<= l 3.8e+78) (fma (* (+ J J) (sinh l)) 1.0 U) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((fma((l * l), 0.3333333333333333, 2.0) * l) * J) * cos((0.5 * K));
	double tmp;
	if (l <= -780.0) {
		tmp = t_0;
	} else if (l <= 0.00035) {
		tmp = (((2.0 * J) * l) * cos((-0.5 * K))) + U;
	} else if (l <= 3.8e+78) {
		tmp = fma(((J + J) * sinh(l)), 1.0, U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J) * cos(Float64(0.5 * K)))
	tmp = 0.0
	if (l <= -780.0)
		tmp = t_0;
	elseif (l <= 0.00035)
		tmp = Float64(Float64(Float64(Float64(2.0 * J) * l) * cos(Float64(-0.5 * K))) + U);
	elseif (l <= 3.8e+78)
		tmp = fma(Float64(Float64(J + J) * sinh(l)), 1.0, U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -780.0], t$95$0, If[LessEqual[l, 0.00035], N[(N[(N[(N[(2.0 * J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 3.8e+78], N[(N[(N[(J + J), $MachinePrecision] * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * 1.0 + U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -780 or 3.7999999999999999e78 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in J around inf

      \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.0%

      \[\leadsto \left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\cos \left(0.5 \cdot K\right)} \]

   if -780 < l < 3.49999999999999996e-4

   1. Initial program 74.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right)} + U \]

   if 3.49999999999999996e-4 < l < 3.7999999999999999e78

   1. Initial program 99.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 99.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 99.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in K around inf

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)}, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(-0.5 \cdot K\right)}, U\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      3. count-2-rev N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(\frac{-1}{2} \cdot K\right), U\right) \]

      4. lower-+.f64 99.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   9. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J + J\right)} \cdot \sinh \ell, \cos \left(-0.5 \cdot K\right), U\right) \]

   10. Taylor expanded in K around 0

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 85.0%

       \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \sinh \ell, 1, U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 76.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.03:\\ \;\;\;\;\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25 + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.03)
   (+ (* (* (* (* K K) l) J) -0.25) U)
   (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.03) {
		tmp = ((((K * K) * l) * J) * -0.25) + U;
	} else {
		tmp = fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.03)
		tmp = Float64(Float64(Float64(Float64(Float64(K * K) * l) * J) * -0.25) + U);
	else
		tmp = fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.03], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision] * -0.25), $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.029999999999999999

   1. Initial program 91.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

   5. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(-0.5 \cdot K\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \left(\frac{-1}{4} \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + \color{blue}{2 \cdot \left(J \cdot \ell\right)}\right) + U \]
   7. Step-by-step derivation

   8. Applied rewrites 43.4%

      \[\leadsto \mathsf{fma}\left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J, \color{blue}{-0.25}, \left(\ell \cdot J\right) \cdot 2\right) + U \]

   9. Taylor expanded in K around inf

      \[\leadsto \frac{-1}{4} \cdot \left(J \cdot \color{blue}{\left({K}^{2} \cdot \ell\right)}\right) + U \]
   10. Step-by-step derivation

   11. Applied rewrites 57.9%

       \[\leadsto \left(\left(\left(K \cdot K\right) \cdot \ell\right) \cdot J\right) \cdot -0.25 + U \]

   if -0.029999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]

   5. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, \color{blue}{J}, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 71.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -205000000000 \lor \neg \left(\ell \leq 0.0265\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -205000000000.0) (not (<= l 0.0265)))
   (* (* (fma (* l l) 0.3333333333333333 2.0) l) J)
   (fma (* 2.0 l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -205000000000.0) || !(l <= 0.0265)) {
		tmp = (fma((l * l), 0.3333333333333333, 2.0) * l) * J;
	} else {
		tmp = fma((2.0 * l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -205000000000.0) || !(l <= 0.0265))
		tmp = Float64(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l) * J);
	else
		tmp = fma(Float64(2.0 * l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -205000000000.0], N[Not[LessEqual[l, 0.0265]], $MachinePrecision]], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.05e11 or 0.0264999999999999993 < l

   1. Initial program 99.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]

   5. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   6. Taylor expanded in J around inf

      \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.3%

      \[\leadsto \left(\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\cos \left(0.5 \cdot K\right)} \]

   9. Taylor expanded in K around 0

      \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 61.7%

       \[\leadsto \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J \]

   if -2.05e11 < l < 0.0264999999999999993

   1. Initial program 74.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lower-fma.f64 74.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      5. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

      9. sinh-undef N/A

         \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

      13. lower-sinh.f64 99.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

      15. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

      18. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

      20. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot 2, \cos \left(-0.5 \cdot K\right), U\right)} \]

   8. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 89.3%

       \[\leadsto \mathsf{fma}\left(\ell \cdot J, \color{blue}{2}, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 89.3%

       \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -205000000000 \lor \neg \left(\ell \leq 0.0265\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.5% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, J, U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma (* (fma (* l l) 0.3333333333333333 2.0) l) J U))

C

double code(double J, double l, double K, double U) {
	return fma((fma((l * l), 0.3333333333333333, 2.0) * l), J, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l), J, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision] * J + U), $MachinePrecision]

Derivation

1. Initial program 87.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]

5. Applied rewrites 86.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

6. Taylor expanded in K around 0

   \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 75.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell, \color{blue}{J}, U\right) \]
9. Add Preprocessing

Alternative 17: 54.0% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 \cdot \ell, J, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma (* 2.0 l) J U))

C

double code(double J, double l, double K, double U) {
	return fma((2.0 * l), J, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(2.0 * l), J, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(2.0 * l), $MachinePrecision] * J + U), $MachinePrecision]

Derivation

1. Initial program 87.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

   3. lower-fma.f64 87.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

   5. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(e^{\ell} - e^{-\ell}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

   6. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \left(\color{blue}{e^{\ell}} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right) \]

   7. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - \color{blue}{e^{-\ell}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

   8. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right), \cos \left(\frac{K}{2}\right), U\right) \]

   9. sinh-undef N/A

      \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right) \cdot \sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot 2\right)} \cdot \sinh \ell, \cos \left(\frac{K}{2}\right), U\right) \]

   13. lower-sinh.f64 100.0

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \color{blue}{\sinh \ell}, \cos \left(\frac{K}{2}\right), U\right) \]

   14. lift-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\frac{K}{2}\right)}, U\right) \]

   15. cos-neg-rev N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

   16. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}, U\right) \]

   17. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right), U\right) \]

   18. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

   19. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}, U\right) \]

   20. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{\color{blue}{-2}}\right), U\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot 2\right) \cdot \sinh \ell, \cos \left(\frac{K}{-2}\right), U\right)} \]

5. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 65.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot 2, \cos \left(-0.5 \cdot K\right), U\right)} \]

8. Taylor expanded in K around 0

   \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
9. Step-by-step derivation

10. Applied rewrites 56.5%

    \[\leadsto \mathsf{fma}\left(\ell \cdot J, \color{blue}{2}, U\right) \]
11. Step-by-step derivation

12. Applied rewrites 56.5%

    \[\leadsto \mathsf{fma}\left(2 \cdot \ell, J, U\right) \]
13. Add Preprocessing

Alternative 18: 36.9% accurate, 330.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, l, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 87.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Applied rewrites 37.6%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025021 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))